The modified boundary contact process: invariant measures and critical extinction

Abstract

We study the one-dimensional modified boundary contact process, in which infections across the two boundary edges of the infected region occur at rate $\hat{I}{»}_e$, while all other infections occur at rate $\hat{I}{»}_i$. We prove two results in different parts of the phase diagram. First, in the non-attractive region $\hat{I}{»}_e>\hat{I}{»}_i\ge \hat{I}{»}_c$, where $\hat{I}{»}_c$ is the critical parameter of the standard contact process, the process seen from its rightmost infected site converges from every semi-infinite initial configuration to an invariant measure. Second, on the critical curve in the attractive region, the infection dies out almost surely.